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mitchell porter
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- TL;DR Summary
- Paper claims to obtain crucial physical quantities from stringy application of pure math. It's probably just "numerology" but let's have a look
What follows may be the bizarrest arxiv paper with a successful calculation, since 2009's "Higgs Boson Mass predicted by the Four Color Theorem". That paper actually got the mass roughly right, using a simple formula motivated by a premise that didn't make sense to anyone else.
The current paper claims to obtain the Higgs boson mass and the cosmological constant, both of which are already measured. But it obtains them by using a construction from one of the realms of string theory that has hitherto been purely of mathematical interest, and does so in a way that seems to be disconnected from any actual physics of the Higgs boson. So I think this will turn out to be just "numerology". But it may be worth trying to understand at least the first steps of the author's thinking...
https://arxiv.org/abs/2309.09762
Monstrous Higgs and Leech Spacetime
Seyed M.F. Khaki
[Submitted on 18 Sep 2023]
The 26-dimensionality of the bosonic string spacetime signals a connection to some advanced and exotic math, in particular the Monster group, a notoriously complicated finite symmetry group of immense complexity. There is a particular highly symmetric 24-dimensional lattice called the Leech lattice (which is of interest e.g. as an error-correcting code), and if 24 dimensions of the bosonic string space-time are compactified on the Leech lattice, you get a vacuum whose string states form representations of the Monster group. That was discovered in the late 1980s and has some relationship to the work of Fields medalist Richard Borcherds.
The next chapter came around 2007 when Edward Witten postulated ("Three-Dimensional Gravity Revisited") that this vacuum is holographically dual to pure gravity in 3-dimensional Anti de Sitter space. The basic equation here is 26-24+1 = 3... start in 26 dimensions, compactify 24 of them so you have an effectively 2-dimensional spacetime, then add one dimension holographically. The reason the dual is purely gravitational has to do with the special geometric properties of the Leech lattice, which have the effect (if I remember correctly) of removing all the spin-1 states of the string (the gauge bosons), leaving only the spin-2 state that produces the graviton (and higher massive states). Witten in his paper did amazing things like obtaining properties of AdS3 black holes by taking the logarithm of representations of the Monster, which end up corresponding to black hole microstates.
Anyway, although our spacetime has 4 dimensions, not 3, and has forces besides gravity, it's tempting to believe that this peculiar relationship between "gravity" and "complete breaking of spacetime symmetries" and "the Monster group" has something to do with physical reality. If I was a loop quantum gravity person, maybe I'd try to build a combinatorial spacetime out of Monster representations... In any case, this brings us to the present paper.
On a superficial reading, I see three things:
1) The author says he is concerned with a dS/CFT dual of the Monster string theory, not AdS/CFT. This means he wants to place the Monster string theory in a timeless state, probably at future infinity, and have the dimension of time emerge holographically. The problem is, this will still only give us a 3-dimensional spacetime. The abstract suggests that spacetime is "globally 3D and locally 4D"; I have no idea how that is supposed to work, but maybe it's explained somewhere.
2) How is the Higgs boson mass obtained? This is on page 4. The key formula is equation 7, and the ingredients are legitimate math (24, the dimension of the Leech lattice; q^3 - q, a formula from finite group theory; ~ 10^53, the number of elements of the Monster group). The question is whether they make sense in this combination. For example, suddenly we are talking about math over a finite field, F_q. It's not entirely clear, but it seems as if the author is using this in place of the continuum to describe space-time. This is a legitimate speculation that has been made in various forms (and there is such a thing as e.g. p-adic string theory), but I don't think it has been a part of Monster string theory before.
The author also makes another offhand comment about how the Virasoro algebra has to be modified a little bit in De Sitter space. It's a bad sign when someone, on the way to deriving a result, says in passing that the usual rules are going to be changed. If this change is a rigorous implication, it should be argued rigorously; if it's a speculation, its speculative character should be highlighted. Otherwise it may just be a handwaving rationalization.
Solving (7) for q just gives a pure number. In order to get a mass, the author takes one final step and decrees that this is to be interpreted as a mass measured in Planck units. And here we finally get something like the Higgs boson mass. Again, the actual logic of this needs to be scrutinized closely. It has to make sense, not just as algebra, but as reasoning about the properties of entities in a physical theory.
3) On the next page, we get to the cosmological constant. It's more numerology that I won't even try to unpick just yet, but this time, we calculate a certain quantity for two sectors of the string theory, twisted and untwisted, and after making them dimensional, we get the dark energy density and the VEV of the Higgs field respectively.
That's all I have time for now, but hopefully it will help perplexed readers get a basic understanding of what's going on in the paper.
The current paper claims to obtain the Higgs boson mass and the cosmological constant, both of which are already measured. But it obtains them by using a construction from one of the realms of string theory that has hitherto been purely of mathematical interest, and does so in a way that seems to be disconnected from any actual physics of the Higgs boson. So I think this will turn out to be just "numerology". But it may be worth trying to understand at least the first steps of the author's thinking...
https://arxiv.org/abs/2309.09762
Monstrous Higgs and Leech Spacetime
Seyed M.F. Khaki
[Submitted on 18 Sep 2023]
So, a few orthodox basics first. The original form of string theory is the bosonic string, it has 26 dimensions and no fermions. The ground state has a tachyon, which in modern physics means, not that it moves faster than light, but that it has negative mass squared, and can therefore be produced in infinite quantities from the vacuum state. The bosonic string vacuum is therefore unstable, and possibly decays immediately to the 10-dimensional superstring that is the one actually used in string phenomenology.
The 26-dimensionality of the bosonic string spacetime signals a connection to some advanced and exotic math, in particular the Monster group, a notoriously complicated finite symmetry group of immense complexity. There is a particular highly symmetric 24-dimensional lattice called the Leech lattice (which is of interest e.g. as an error-correcting code), and if 24 dimensions of the bosonic string space-time are compactified on the Leech lattice, you get a vacuum whose string states form representations of the Monster group. That was discovered in the late 1980s and has some relationship to the work of Fields medalist Richard Borcherds.
The next chapter came around 2007 when Edward Witten postulated ("Three-Dimensional Gravity Revisited") that this vacuum is holographically dual to pure gravity in 3-dimensional Anti de Sitter space. The basic equation here is 26-24+1 = 3... start in 26 dimensions, compactify 24 of them so you have an effectively 2-dimensional spacetime, then add one dimension holographically. The reason the dual is purely gravitational has to do with the special geometric properties of the Leech lattice, which have the effect (if I remember correctly) of removing all the spin-1 states of the string (the gauge bosons), leaving only the spin-2 state that produces the graviton (and higher massive states). Witten in his paper did amazing things like obtaining properties of AdS3 black holes by taking the logarithm of representations of the Monster, which end up corresponding to black hole microstates.
Anyway, although our spacetime has 4 dimensions, not 3, and has forces besides gravity, it's tempting to believe that this peculiar relationship between "gravity" and "complete breaking of spacetime symmetries" and "the Monster group" has something to do with physical reality. If I was a loop quantum gravity person, maybe I'd try to build a combinatorial spacetime out of Monster representations... In any case, this brings us to the present paper.
On a superficial reading, I see three things:
1) The author says he is concerned with a dS/CFT dual of the Monster string theory, not AdS/CFT. This means he wants to place the Monster string theory in a timeless state, probably at future infinity, and have the dimension of time emerge holographically. The problem is, this will still only give us a 3-dimensional spacetime. The abstract suggests that spacetime is "globally 3D and locally 4D"; I have no idea how that is supposed to work, but maybe it's explained somewhere.
2) How is the Higgs boson mass obtained? This is on page 4. The key formula is equation 7, and the ingredients are legitimate math (24, the dimension of the Leech lattice; q^3 - q, a formula from finite group theory; ~ 10^53, the number of elements of the Monster group). The question is whether they make sense in this combination. For example, suddenly we are talking about math over a finite field, F_q. It's not entirely clear, but it seems as if the author is using this in place of the continuum to describe space-time. This is a legitimate speculation that has been made in various forms (and there is such a thing as e.g. p-adic string theory), but I don't think it has been a part of Monster string theory before.
The author also makes another offhand comment about how the Virasoro algebra has to be modified a little bit in De Sitter space. It's a bad sign when someone, on the way to deriving a result, says in passing that the usual rules are going to be changed. If this change is a rigorous implication, it should be argued rigorously; if it's a speculation, its speculative character should be highlighted. Otherwise it may just be a handwaving rationalization.
Solving (7) for q just gives a pure number. In order to get a mass, the author takes one final step and decrees that this is to be interpreted as a mass measured in Planck units. And here we finally get something like the Higgs boson mass. Again, the actual logic of this needs to be scrutinized closely. It has to make sense, not just as algebra, but as reasoning about the properties of entities in a physical theory.
3) On the next page, we get to the cosmological constant. It's more numerology that I won't even try to unpick just yet, but this time, we calculate a certain quantity for two sectors of the string theory, twisted and untwisted, and after making them dimensional, we get the dark energy density and the VEV of the Higgs field respectively.
That's all I have time for now, but hopefully it will help perplexed readers get a basic understanding of what's going on in the paper.